### New Results on $k$-Independence of Graphs

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Let $G = (V, E)$ be a graph and $k \geq 0$ an integer. A $k$-independent set $S \subseteq G$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. Denote by $\alpha_{k}(G)$ the maximum cardinality of a $k$-independent set of $G$. For a graph $G$ on $n$ vertices and average degree $d$, Turán's theorem asserts that $\alpha_{0}(G) \geq \frac{n}{d+1}$, where the equality holds if and only if $G$ is a union of cliques of equal size. For general $k$ we prove that $\alpha_{k}(G) \geq \dfrac{(k+1)n}{d+k+1}$, improving on the previous best bound $\alpha_{k}(G) \geq \dfrac{(k+1)n}{ \lceil d \rceil+k+1}$ of Caro and Hansberg [E-JC, 2013]. For $1$-independence we prove that equality holds if and only if $G$ is either an independent set or a union of almost-cliques of equal size (an almost-clique is a clique on an even number of vertices minus a $1$-factor). For $2$-independence, we prove that equality holds if and only if $G$ is an independent set. Furthermore when $d>0$ is an integer divisible by 3 we prove that $\alpha_2(G) \geq \dfrac{3n}{d+3} \left( 1 + \dfrac{12}{5d^2 + 25d + 18} \right)$.